(WIP) Discrete logarithm #110
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This is a first implementation of a discreteLogarithm function. It uses a naive algorithm that is really slow on large moduli. In subsequent commits I will add a benchmark and use a more sophisticated algorithm.
Currently this takes 1.22 seconds on my laptop! Lots of room for improvement
Benchmark gives a result of 7ms, a speed-up of about 175 times!
This test currently fails and needs to be resolved.
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@Taneb just a gentle reminder that you intended to resolve remaining issues. Or we can merge as is and restrict inputs of discrete logarithm to primitive roots only. |
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Sorry, I've been busier than I expected! 😅 Going to do some work on this tonight, though |
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I think we can introduce a newtype and amend the signature of isPrimitiveRoot
:: KnownNat n => Mod n -> Maybe (PrimitiveRoot n)Consequently, discreteLogarithm :: (KnownNat m, Integral b) => Mod m -> PrimitiveRoot m -> Maybe bThis eliminates the current issue with logarithms by non-primitive base (Pollard's algorithm supposes that the base is primitive) without compromising performance or correctness. |
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Oooh, that's a good idea. I'll implement that this evening |
…' to use this. I'm not sure about the name for primitiveRootGetMod. Another thing I thought of is it might be nice to have a type for members of the multiplicative group modulo m, that is, the subset of Mod m coprime to m.
| -- | 'PrimitiveRoot n' is a type which is only inhabited by primitive roots of | ||
| -- n. | ||
| newtype PrimitiveRoot n = PrimitiveRoot (Mod n) | ||
| deriving (Eq, Ord, Show) |
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Let us define newtype PrimitiveRoot n = PrimitiveRoot { unprimitiveRoot :: Mod n } and export the type and the accessor, but not the constructor.
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@Taneb any chance to finish it soon? |
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Closing, superseded by #130. Anyway, thanks for you efforts. |
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Sorry I neglected this issue! I'm glad it's been solved by someone, I had... less time and headspace for this than I'd have liked |
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It was a pleasure to work with you at ZuriHac. |
This is the work I started at ZuriHac. It implements Pollard's rho algorithm to compute the discrete logarithm. This has the deficiency that there are some solutions it cannot find, such as 2^x == 7 (mod 9), which I would like to resolve before this PR gets merged.